= Solution
The <Gauss map> sends each point of an oriented surface to its chosen unit normal $N(p)$. Both $T_pS$ and $T_{N(p)}S^2$ are the plane orthogonal to $N(p)$, so the derivative is an endomorphism of that plane. In local coordinates $r(u,v)$, differentiating $\langle N,r_j\rangle=0$ gives $\langle N_i,r_j\rangle=-\langle N,r_{ij}\rangle$. Since $r_{ij}=r_{ji}$, this equals $\langle N_j,r_i\rangle$. Bilinearity then gives $\langle dN(v),w\rangle=\langle v,dN(w)\rangle$ for all tangent vectors. \b[The derivative of the Gauss map is self-adjoint.]
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